It is well known that splines play an important role in many 0elds, especially, their close relationship with wavelets makes them have more widespread applications in numerous scienti0c and engineering domains. Univariate and bivariate splines have been well studied and lots of results have been obtained. Because of the intrinsic diffculty between bivariate case and higher-dimension (three or more dimensions) settings, the study of splines on higher dimensions are very limited. For example, the study of the bivariate splines on a three-direction mesh triangulation has obtained many important and excellent results, but almost all of those results have no analog generalization to higher dimensions. In this paper, we will study the higher-dimension splines de0ned on n+1 mesh simplical partitions which is the analog of bivariate splines on three-mesh triangulations. We have also pointed out many interesting di6erences between bivariate splines and higher-dimensional cases. Our main results are that, similar to bivariate and trivariate cases, a necessary and suffcient condition for Sr k (△) to contain a B-spline is k≥1 2(r+1) (n+1) for r≥1 being odd and k≥1 2r(n+1)+1 for r≥0 being even.